C₃×#9068 #10627675

order	768
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#9068

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#120
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#386
prime radical	C₂³×C₃	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

404 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(5,9,11,14)(6,12,13,15)(20,21,24,28,25,23,32,27)(22,26,30,34,31,29,35,33);(4,5,7,14,8,11,16,9)(6,10,15,18,13,19,12,17)(20,22,24,30,25,31,32,35)(21,26,28,34,23,29,27,33);(4,6,8,13)(5,10,11,19)(7,15,16,12)(9,17,14,18)(20,23)(21,25)(22,29)(24,27)(26,31)(28,32)(30,33)(34,35)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2056);